Closeness of the Solutions of Approximately Decoupled Damped Linear Systems to Their Exact Solutions

1992 ◽  
Vol 114 (3) ◽  
pp. 369-374 ◽  
Author(s):  
S. M. Shahruz ◽  
G. Langari
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Linear Systems ◽  
Equations Of Motion ◽  
Second Order ◽  
Order System ◽  
System Response ◽  
Damping Matrix

The simplest technique of decoupling the normalized equations of motion of a linear nonclassically damped second-order system is to neglect the off-diagonal elements of the normalized damping matrix. In this paper, the error introduced in the system response due to this decoupling technique is studied. Conditions are derived under which the solution of an approximately decoupled system is “close” to the exact solution of the system.

Approximate Decoupling of Weakly Nonclassically Damped Linear Second-Order Systems Under Harmonic Excitations

1993 ◽  
Vol 115 (1) ◽  
pp. 214-218 ◽  
Author(s):  
S. M. Shahruz ◽  
A. K. Packard
Keyword(s):  
Equations Of Motion ◽  
Excitation Frequency ◽  
Approximation Error ◽  
Second Order ◽  
Order System ◽  
Damping Matrix ◽  
Approximation Errors ◽  
Harmonic Excitations ◽  
Second Order Systems ◽  
Damped Systems

A simple and commonly used approximate technique of solving the normalized equations of motion of a nonclassically damped linear second-order system is to decouple the system equations by neglecting the off-diagonal elements of the normalized damping matrix, and then solve the decoupled equations. This approximate technique can result in a solution with large errors, even when the off-diagonal elements of the normalized damping matrix are small. Large approximation errors can arise in lightly damped systems under harmonic excitations when some of the undamped natural frequencies of the system are close to the excitation frequency. In this paper, a rigorous analysis of the approximation error in lightly damped systems is given. Easy-to-check conditions under which neglecting the off-diagonal elements of the normalized damping matrix can result in large approximation errors are presented.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Evaluation of Parametric Vibration and Stability of Flexible Cam-Follower Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 291-297 ◽  
Author(s):  
A. I. Mahyuddin ◽  
A. Midha ◽  
A. K. Bajaj
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Valve Train ◽  
Time Dependent ◽  
Equivalent Model ◽  
System Response ◽  
Parametric Stability ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Cam Follower

A method to study parametric stability of flexible cam-follower systems is developed. This method is applied to an automotive valve train which is modeled as a single-degree-of-freedom vibration system. The inclusion of the transverse and rotational flexibilities of the camshaft results in a system governed by a second-order, linear, ordinary differential equation with time-dependent coefficients. This class of equations, known as Hill’s equations, merits special notice in determination of the system response and stability. The analysis includes development of the equivalent model of the system, derivation of its equation of motion, and a method to evaluate its parametric stability based on Floquet theory. A closed-form numerical algorithm, developed to compute the periodic response of systems governed by second-order, linear, ordinary differential equations of motion with time-dependent coefficients, is utilized. The results of this study are presented in a companion paper in the forms of parametric stability charts and three-dimensional stability and response charts.

Analytical Calculation of the Position Loop Gain for Linear Motor CNC Machine Tool

2012 ◽  
Vol 186 ◽  
pp. 182-187 ◽  
Author(s):  
Zoran Pandilov ◽  
Vladimir Dukovski
Keyword(s):  
Machine Tool ◽  
Linear Motor ◽  
Second Order ◽  
Order System ◽  
System Response ◽  
Loop Gain ◽  
Cnc Machine ◽  
Servo Drive ◽  
Simple Method ◽  
Following Error

One of the most important factors which influence on the dynamical behavior of the linear motor servo drives for CNC machine tools is position loop gain or Kv factor. From the magnitude of the Kv-factor depends tracking or following error. In multi-axis contouring the following errors along the different axes may cause form deviations of the machined contours. Generally position loop gain Kv should be high for faster system response and higher accuracy, but the maximum gains allowable are limited due to undesirable oscillatory responses at high gains and low damping factor. Usually Kv factor is experimentally tuned on the already assembled machine tool. This paper presents a simple method for analytically calculation of the position loop gain Kv. A combined digital-analog model of the 4-th order of the position loop is presented. In order to ease the calculation, the 4-th order system is simplified with a second order model. With this approach it is very easy to calculate the Kv factor for necessary position loop damping. The difference of the replacement of the 4-th order system with second order system is presented with the simulation program MATLAB. Analytically calculated Kv factor is function of the nominal angular frequency  and damping D of the linear motor servo drive electrical parts (motor and regulator), as well as sampling period T. :The influence of nonlinearities was taken with the correction factor. Our investigations have proven that experimentally tuned Kv factor differs from analytically calculated Kv factor less than 10%, which is completely acceptable

scholarly journals Forensic Engineering Analysis Of Trailer Sway Accidents

2008 ◽  
Vol 25 (1) ◽  
Author(s):  
Mark A.M. Ezra ◽  
Landiss Danel J.
Keyword(s):  
Data Reduction ◽  
Damping Ratio ◽  
Experimental System ◽  
Second Order ◽  
Coupled Systems ◽  
Order System ◽  
System Response ◽  
Statistical Validation ◽  
Response Data ◽  
Data Reduction Method

This Paper Describes A Method For The Reduction Of System Response Data For Second Order Electrical Or Mechanical Systems When That Data Is Available Only In Graphical Format. The Method Of Data Reduction Described Allows Quantitative Evaluation Of Generally Accepted Second Order System Parameters Such As: System Time Constant, Damped Natural Frequency, Damping Ratio, And Exponential Decay Time. The Discussion Includes The Application Of The Described Graphical Technique To Experimental System Response Data Of Coupled Systems, But Whose Experimental Response Approximates That Of An Isolated Second Order System. A Practical Application Of The Described Data Reduction Method Is Covered In Detail. The Described Technique Is Applied To The Analysis Of Data Obtained Experimentally For The Response Of A Tow Vehicle And Trailer System To A Standardized Steering Disturbance. Finally, The Statistical Validation For Experimental System Response Data And The Results Obtained From The Analysis Of Such Data, Using The Described Graphical Method, Is Discussed.

Simulation and Nonlinear Dynamics Analysis of Planing Hulls

1995 ◽  
Vol 117 (1) ◽  
pp. 38-45 ◽  
Author(s):  
J. D. Hicks ◽  
A. W. Troesch ◽  
C. Jiang
Keyword(s):  
Differential Equations ◽  
System Analysis ◽  
Equations Of Motion ◽  
Path Following ◽  
Experimental Tests ◽  
Second Order ◽  
Design Tool ◽  
Continuation Methods ◽  
System Response ◽  
Analytic Expressions

The high speeds, small trim angles, and shallow drafts of planing hulls produce large changes in vessel wetted surface which, in turn, lead to significant hydrodynamic and dynamic nonlinearities. Due to the complex nonlinearities of this type of craft, naval architects and planing boat designers tend to rely upon experimental tests or simulation for guidance. In order for simulation to be an effective design tool, a fundamental understanding of the system’s dynamic characteristics is required. This paper describes a developing methodology by which the necessary insight may be obtained. A demonstration of the combined use of modern methods of dynamical system analysis with simulation is given in the evaluation of the vertical motions of a typical planing hull. Extending the work of Troesch and Hicks (1992) and Troesch and Falzarano (1993), the complete nonlinear hydrodynamic force and moment equations of Zarnick (1978) are expanded in a multi-variable Taylor series. As a result, the nonlinear integro-differential equations of motion are replaced by a set of highly coupled, ordinary differential equations with constant coefficients, valid through third order. Closed-form, analytic expressions are available for the coefficients (Hicks, 1993). Numerical examples for all first-order and some second-order terms are presented. Once completely determined, the coefficient matrices will serve as input to path following or continuation methods (e.g., Seydel, 1988) where heave and pitch magnification curves can be generated, allowing the entire system response to be viewed. The branching behavior of the solutions resulting from a variation of the center of gravity is examined in detail. These studies of the second-order accurate model show the potential of the method to identify areas of critical dynamic response, which in turn can be verified and explored further through the use of the simulator.

scholarly journals On Decoupling the Equations of Motion of Nonclassically Damped Systems

1989 ◽  
Author(s):  
F. Ma ◽  
J. H. Hwang
Keyword(s):  
Approximate Solution ◽  
Linear Systems ◽  
Error Bounds ◽  
Equations Of Motion ◽  
Effective Procedure ◽  
Common Procedure ◽  
Damping Matrix ◽  
Alternative Techniques ◽  
Damped Systems

One common procedure in the solution of a damped linear systems with small off-diagonal damping elements is to neglect the off-diagonal elements of the normalized damping matrix. The extent of approximation introduced by this method of decoupling the system is evaluated, and tight error bounds are derived by alternative techniques. An effective procedure to improve the accuracy of the approximate solution is outlined.

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